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Mirrors > Home > MPE Home > Th. List > Mathboxes > psubclssatN | Structured version Visualization version GIF version |
Description: A closed projective subspace is a set of atoms. (Contributed by NM, 25-Jan-2012.) (New usage is discouraged.) |
Ref | Expression |
---|---|
psubclssat.a | ⊢ 𝐴 = (Atoms‘𝐾) |
psubclssat.c | ⊢ 𝐶 = (PSubCl‘𝐾) |
Ref | Expression |
---|---|
psubclssatN | ⊢ ((𝐾 ∈ 𝐷 ∧ 𝑋 ∈ 𝐶) → 𝑋 ⊆ 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | psubclssat.a | . . 3 ⊢ 𝐴 = (Atoms‘𝐾) | |
2 | eqid 2737 | . . 3 ⊢ (⊥𝑃‘𝐾) = (⊥𝑃‘𝐾) | |
3 | psubclssat.c | . . 3 ⊢ 𝐶 = (PSubCl‘𝐾) | |
4 | 1, 2, 3 | psubcliN 37689 | . 2 ⊢ ((𝐾 ∈ 𝐷 ∧ 𝑋 ∈ 𝐶) → (𝑋 ⊆ 𝐴 ∧ ((⊥𝑃‘𝐾)‘((⊥𝑃‘𝐾)‘𝑋)) = 𝑋)) |
5 | 4 | simpld 498 | 1 ⊢ ((𝐾 ∈ 𝐷 ∧ 𝑋 ∈ 𝐶) → 𝑋 ⊆ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1543 ∈ wcel 2110 ⊆ wss 3866 ‘cfv 6380 Atomscatm 37014 ⊥𝑃cpolN 37653 PSubClcpscN 37685 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2708 ax-sep 5192 ax-nul 5199 ax-pow 5258 ax-pr 5322 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2886 df-ral 3066 df-rex 3067 df-rab 3070 df-v 3410 df-dif 3869 df-un 3871 df-in 3873 df-ss 3883 df-nul 4238 df-if 4440 df-pw 4515 df-sn 4542 df-pr 4544 df-op 4548 df-uni 4820 df-br 5054 df-opab 5116 df-mpt 5136 df-id 5455 df-xp 5557 df-rel 5558 df-cnv 5559 df-co 5560 df-dm 5561 df-iota 6338 df-fun 6382 df-fv 6388 df-psubclN 37686 |
This theorem is referenced by: pmapidclN 37693 psubclinN 37699 paddatclN 37700 pclfinclN 37701 poml6N 37706 osumcllem3N 37709 osumcllem9N 37715 osumcllem11N 37717 osumclN 37718 |
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